What is the method to obtain this integral result?

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In summary, the conversation discusses a result involving an integral with a complicated integrand and the question of how it was derived. One participant suggests using the binomial expansion, but the result is different. Another participant suggests using the derivative of the CDF of a variable called gamma, but the person is still stuck and asks for further hints.
  • #1
EngWiPy
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Hello,

I am reading some material that using mathematics extensively, and I encountered with the following result:

[tex]\frac{N}{\overline{\gamma}}\,\int_0^{\infty}\gamma\,\left[1-\mbox{e}^{-\gamma/\overline{\gamma}}\right]^{N-1}\,\mbox{e}^{-\gamma/\overline{\gamma}}\,d\gamma=\,\overline{\gamma}\sum_{k=1}^N\frac{1}{k}[/tex]

How did they get there? I tried to use the binomial expansion and assemble the exponentials, but the result was totally different. Any hint will be highly appreciated.

Thanks in advance
 
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  • #2
Note that the integrand looks very much like a derivative of
[tex]\left( 1 - e^{-\gamma / \overline{\gamma}} \right)^N[/tex]
with respect to either gamma or gamma-bar.
Maybe you can use that to your advantage.
 
  • #3
CompuChip said:
Note that the integrand looks very much like a derivative of
[tex]\left( 1 - e^{-\gamma / \overline{\gamma}} \right)^N[/tex]
with respect to either gamma or gamma-bar.
Maybe you can use that to your advantage.

Yes, you are right. the term [tex]\left[1-\mbox{e}^{-\gamma/\overline{\gamma}}\right]^N[/tex] is the CDF of [tex]\gamma[/tex]. In the integral it is intended to find the statistical average (the expected value) of [tex]\gamma[/tex] which needs the PDF of [tex]\gamma[/tex] which is the derivative of the CDF with respect to [tex]\gamma[/tex]. But I am still stuck. Any further hint?

Thanks in advance
 

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